Renormalization Group in Curved Space and the Problem of Conformal Anomaly
|
|
- Bruce Hutchinson
- 5 years ago
- Views:
Transcription
1 086 Brazilian Journal of Physics, vol 35, no 4B, December, 005 Renormalization Group in Curved Space and the Problem of Conformal Anomaly M Asorey, Departamento de Física Teorica, Universidad de Zaragoza, 50009, Zaragoza, Spain E V Gorbar, Department of Applied Mathematics, University of Western Ontario, London, Ontario 6A 5B7, Canada and I L Shapiro Departamento de Física, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP: , MG, Brazil Received on October, 005 Renormalization Group RG is a powerful method for investigating quantum effects of matter fields on curved background The formalism of RG in curved space is well known since 984, but its applications to cosmology and black hole physics require more knowledge and opens a new interesting field of study We review recent results about the derivation of renormalization group in a mass-dependent scheme and also consider ambiguities of conformal anomaly using dimensional and covariant Pauli-Villars regularizations The Effective Action of vacuum EA is a most useful object to parametrize the quantum effects within quantum field theory in curved space-time [, Sometimes this approach is refereed to as semiclassical approximation, in order to emphasize that the metric is not quantized The EA is a metricdependent functional, at one-loop level it consists of the classical action of vacuum and the contributions of closed loops of free matter fields At higher loops the fields interactions become relevant However, the practical derivations of the EA are mainly restricted by the one-loop order and here we shall also work in the framework of this approximation In d = 4 even the contributions of free matter fields are very complicated to calculate, and this task has been successful only for some particular cases The word particular corresponds to the choice of both the model for the quantum field and the one for the background classical metric The most remarkable example is the case of massless conformal invariant fields on the cosmological homogeneous and isotropic metric background, where the EA can be derived exactly [3, 4 This result has been obtained through integration of conformal anomaly [5, 6, showing the importance of anomaly and the related renormalization group for the analysis of the EA The investigation of EA in more complicated situations, in particular for massive fields, requires a choice of the approximation and of the calculational scheme In this respect the seminal role belongs to the Renormalization Group RG, which enables one to reduce the information about complicated non-local quantum contributions to the relatively simple dependence on a single parameter µ The most simple and most formal version of RG, based on the Minimal Subtraction MS renormalization scheme, has been successfully applied to the quantum field theory in curved space see, eg [ for the On leave from Bogolyubov Institute for Theoretical Physics, 0343 Kiev, Ukraine Also at Tomsk State Pedagogical University, Russia review and introduction The weak side of the MS-based RG is the problem of interpretation of µ and related difficulties with applying RG, eg in cosmology An alternative way is to use some more sophisticated scheme of renormalization, which would not be universal one as the MS-based RG is, but which should be designed especially for the given application Unfortunately, there is no covariant renormalization scheme except the MS one Furthermore, the known renormalization schemes eg the momentum subtraction scheme of renormalization are not really appropriate for the use in the gravitational framework Still the momentum subtraction scheme enables one to extract some relevant information, part of it will be reviewed below The main purpose of the original papers on the subject was to investigate the phenomenon of decoupling of massive fields at low energies for the free fields of different spin [7, the theory with Spontaneous Symmetry Breaking [8, for one particular example of interacting theory [9 and also to clarify some long-standing problem concerning the conformal anomaly [0 The main goal of the works [7 9 was to formulate the effective approach for quantum field theory in curved spacetime The notion of decoupling is an important aspect of effective approach At classical level decoupling means that a heavy field doesn t propagate at low energies This can be easily seen observing the propagator of a massive particle at low energy k + M M + O k M 4, k M The decoupling theorem explains how the suppression of the effects of heavy particles occurs at quantum level Let us start from the pedagogical example and consider QED in flat space The -loop vacuum polarization is e θ µν π 0 dxx x ln m e + p x x 4πµ, where θ µν = p µ p ν p g µν and µ is the parameter of dimensional regularization In the MS renormalization scheme the
2 M Asorey et al 087 β-function β MS results from acting e µ dµ d of θ µν on the form factor β MS e = e3 π 3 This β-function can not tell us about the decoupling In order to define physical β-function, let us apply the mass-dependent renormalization scheme Subtracting the divergence at p = M and taking derivative e M dm d we arrive at β e = e3 π 0 M x x dxx x m e + M x x The UV limit M m e gives β e = β MS e and in the IR limit M m e we meet a quadratic decoupling compared to the UV β-function Appelquist & Carazzone theorem [ β e = e3 60π M m e M 4 + O m 4 4 In order to derive decoupling theorem for gravity we have considered massive fields on the classical metric background [7 As we have already mentioned above, there is no completely covariant technique compatible with the massdependent renormalization schemes We have performed calculations within the linearized gravity on the flat background g µν = η µν + h µν The corrections to the graviton propagator come from the standard Feynman diagrams [7 or may be derived using the covariant heat-kernel solution in the second order in curvature approximation [, 3 The polarization operator must be compared to the tensor structure of the vacuum Lagrangian L = 6πG R + Λ + a C + a E + a 3 R + a 4 R Here C = Cµναβ is the square of the Weyl tensor and E is the integrand of the Gauss-Bonnet topological invariant Euler characteristic E = R µναβ 4R µν + R In the case of massive scalar we meet, after performing all the integrations, the following EA: Γ s = d 4 x { g m 4 4π ε [ C µναβ 60ε + k W C µναβ + R + ξm R ε + [ ξ ε + k R } R 5 where ε = w + ln 4πµ, ξ = ξ 6 The formfactors for the higher derivative terms are m k W = k W a = 8A 5a a + k R = k R a = A ξ + + A 9a 4 3A 8 50, + A 3a ξ A 8a + A a 7 60, 6 where we used notations A = + a ln a + a, a = 4 4m Obviously, constant formfactors mean zero β-functions for Λ and G The situation for the higher derivative terms is quite different For the Weyl term the β-function is β = 4π 8a In the UV and IR limits we have and β IR 80 a 4 6a 4 β UV = 4π 0 + O m p p = 6804π m + O p 4 m 4 A 7, 8 The last formula is nothing but the Appelquist & Carazzone theorem for gravity Here p is the momentum of the linearized gravitational field h µν on flat background For the R term the situation is very similar The bulky expressions for the β-function can be found in [7, the comparison of the UV and IR limits manifests standard quadratic decoupling β UV = 4π ξ + O m p, β IR = p /m 4π [ ξ ξ O p 4 m 4 9 Similar results for the decoupling take place for the massive fermion and massive scalar cases [7 The difference with the contributions of massive scalar consists in the coefficients of the same a-dependent terms One important consequence of this difference can be observed in the theory with broken supersymmetry If we assume that all particles of the MSM are very light compared to the s-particles, we arrive at the plot of the β -function for the local R -term see Fig At this plot the value a = corresponds to the UV limit and the value a = 0 corresponds to the IR limit One can see that the sign of this β-function is changing on the way from UV to IR This change of sign leads to the transition from stable to unstable regimes [4 in the model of anomaly-induced inflation [5 In the theory with Spontaneous Symmetry Breaking SSB the situation is essentially more complicated In this case, even at the tree level one has an infinite number of non-local terms in both vacuum and induced actions of gravity, and all those non-local terms in the vacuum action get renormalized at the quantum level However, the standard decoupling law for the higher derivative terms holds in this case too [8 An expansion g µν = η µν + h µν works well for higher derivative terms, but not for Λ and G Why did we obtain β Λ = β /G 0 in the momentum-subtraction scheme? Let us remind that the same β-functions are non-zero in the MS-renormalization scheme [ In fact, the running in the momentum-subtraction scheme means the presence of a
3 088 Brazilian Journal of Physics, vol 35, no 4B, December, FIG : The plot β a demonstrating the change of sign between UV and IR f = ln /µ -like formfactor In QED, in the UV limit we meet the term e 4 F µνf µν + e4 34π F µν ln µ F µν Similarly in gravity it is possible to insert C µναβ f C µναβ or R f R in the higher derivative sector However, no insertion is possible for Λ and /G, since Λ = 0 and R is a total derivative Does it mean that β Λ and β /G really equal zero? From our point of view the answer is negative, for otherwise we meet a divergence between the mass-dependent renormalization scheme and MS-scheme in the UV where they are supposed to be the same Perhaps calculations on a flat background are not appropriate for deriving the renormalization group equations for Λ and /G This hypothesis is quite natural, especially because flat space is not a classical solution in the presence of the cosmological constant Let us notice that the practical derivation of the decoupling law for Λ and /G may have very interesting cosmological and astrophysical applications [6 One of the applications of our expression for the effective action of massive fields 5 is the new way of analyzing the conformal anomaly a typical phenomenon for the quantized massless fields on classical curved background [7 As an example, consider scalar field L = { ϕ + m ϕ + ξ + 6 Rϕ } In the m = 0, ξ = /6 limit of 5 we obtain, in the R sector, 804π d 4 xg / R 0 Due to the identity g µν g g µν d 4 x gr = R < T µ µ >= 3604π [ 3C E + R, where C = C 4 = R µναβ R µν + /3R is the square of the result 0 provides a perfect fit with the R term in the local means with respect to local conformal symmetry conformal anomaly obtained by the point-splitting method [8, ζ-regularization [9 and other methods see [ for the review the Weyl tensor at n = 4 and E = R µναβ 4R µν + R is the integrand of the Gauss-Bonnet invariant We added these two terms for consistency, despite our main attention will be paid to the R term Let us notice that the above result has been obtained via the massless conformal limit in the massive and non-conformal generally, ξ /6 theory Indeed, for any ξ /6, the R term is subject of infinite renormalization procedure and therefore the above limit is ambiguous Indeed, this situation may look quite trivial, because the term of our interest is local However, a relevant detail is that, in the conformal theory, the counterterm of R type is not needed, hence one can thing that the ambiguity is nothing but a consequence of a wrong choice of regularization scheme The anomaly can be seen as a consequence of an infinite regularization of the higher derivative terms in the vacuum action The simplest way to derive the conformal anomaly is using the dimensional regularization [6, however this calculation meets certain problem The usual scheme of calculation [6 is as follows The renormalized -loop EA is a sum of three parts Γ ren = S vac + Γ + S vac, where S vac is the classical vacuum action Indeed, this action can be chosen to be conformal invariant for the case of conformal matter [0 Furthermore, Γ is the non-renormalized quantum correction, which is divergent, conformal invariant as far as the matter field was formulated conformal invariant in n 4 and non-local Finally, S vac = 3C 4π E + R n 4 is the local counterterm which is not conformal invariant is n 4 Obviously, the anomaly comes from the unique nonconformal component, that is S vac T =< T µ µ >= g g µν S vac g µν For the C and E terms the application of this formula leads us, unambiguously, to the expression Let us now consider the remaining term and see that in this case the situation is more complicated Indeed g µν d n x g R 0 This shows that, in the dimensional regularization, the Rtype counterterm does not contribute to the R term in the
4 M Asorey et al 089 anomaly Of course, this is not a natural output, especially because the anomaly for the global conformal symmetry does depend on all counterterms Furthermore, other regularization schemes also link the two coefficients in a direct way, so here we meet a divergence between the results obtained in distinct regularizations According to [6, the R term in the anomaly is coming from the C -type counterterm via the identity g µν g µν d n x g C 4 n 4 = C R 3 n 4 3 Hence T = /3β Weyl R + For the scalar and spinor fields this relation gives the same result as other regularization schemes However, for the vector field and also for the higher derivative scalar [3 and fermion [ the coefficients are different Hence it is reasonable to look again at the definition 3 Why we have to choose S vac = gc 4? In fact, if we choose the square of the Weyl tensor in an arbitrary dimension d, C d = R µναβ 4 d R µν + d d R, 4 where d = 4 + On 4, the corresponding counterterm will be perfectly consistent with its destination, for it is local and cancels the divergence in the one-loop EA However the result for the R term in the anomaly essentially depends on the choice of d Eg for C n there is no R term at all and for the d = n + γ [n 4 the R term is proportional to an arbitrary parameter γ We can conclude that the dimensional regularization does not predict the coefficient of the R term in the anomaly Let us notice that the change of parameter γ in the counterterm which we add to Γ by hands is completely equivalent to adding a finite R term to the classical action Adding this term is absolutely safe procedure, because it is not a subject of infinite regularization From the point of view of renormalizability this term is not necessary but it is not forbidden either Hence the arbitrariness we have found in the dimensional regularization is nothing else but the consequence of the identity This arbitrariness takes place also for other fields and finally the coefficient α in the general expression for the anomaly < T µ µ >= [ wc + be + α R 5 can not be defined within the dimensional regularization We remark that this arbitrariness is exactly the same we already met when taking the massless conformal limit in the effective action of a massive scalar theory with an arbitrary ξ In order to complete the story we can answer the following question: is it possible to observe an arbitrariness in α in some other regularization? In order to address this issue we need another example of covariant regularization which should admit certain freedom in choosing regularization parameters And our result for the effective action 5 enables us to build the regularization of this sort Consider covariant version of the Pauli-Villars regularization, known from Yang-Mills and Chern-Simons theories [3, 4 The Pauli-Villars regularization implies introducing massive auxiliary fields with distinct Grassmann parity which cancel all quartic, quadratic, log divergences in a covariant way S reg = i=0 d 4 x g { ϕ i + ξ i R + m i ϕ i } The physical scalar field ϕ ϕ 0 is conformal ξ = /6, m 0 = 0 and bosonic s 0 = On the other hand, the Pauli-Villars regulators ϕ i i =,, are massive m i = µ i M 0 and can have either bosonic s i = or fermionic statistics s i = Since they are not observed, there is no violation of the spin-statistics theorem here After UV limit M we arrive at the vacuum EA The calculation is based on our result 5, as we already mentioned We assume that the Pauli-Villars regulators might have non-conformal couplings ξ i 6 The regularized effective action is Γ reg = lim Λ i=0 s i Γ i m i,ξ i,λ, where Λ is an auxiliary momentum cut-off All divergences cancel out due to the Pauli-Villars conditions, which have, in our case, the following form: i= i= s i µ i = i= s i = ; i= s i µ 4 i = quartic divs s i ξ i = 0; quadratic 6 i= s i ξ i 6 = 0; log A simple solution of these equations is s,,3,4,5 =,4,,,, µ,,3,4,5 = 4,3,,3,4, ξ i = µ i + /6 Alternatively we can choose all ξ i = /6, so there is an arbitrariness here The conformal anomaly in the covariant Pauli-Villars regularization is given by [ T = 4π 80 E 0 C + R, 80 where = i= s i ξ i 6 ln µ i,
5 090 Brazilian Journal of Physics, vol 35, no 4B, December, 005 Again, we meet an ambiguity It is easy to see that, exactly as in the dimensional regularization case, the ambiguity is related to the freedom to add the finite gr -term to the classical vacuum action The only way to fix the ambiguity in α is through a special renormalization condition for the gr -term In the free scalar theory this looks as an additional complication, because this term is not renormalized But as we know from the literature [, in the interacting theory the gr -type counterterm emerges anyway at higher loops and, therefore, this extra condition does not lead to an essential modification of the theory structure At that point one has to remember that the dynamical effect comes from the gr -term, and not from the total derivative g R Looking from this perspective, we can see that the renormalization condition for the gr -term is fixing the initial point of the renormalization group trajectory for the corresponding coupling, and not the shape of the trajectory In other words, the curve shown at the Figure remains valid if we impose the renormalization condition for the gr -term in a proper way As a result the model of anomaly-induced inflation is not jeopardized by the α ambiguity in the conformal anomaly In conclusion, we have established the background of an effective approach to QFT in curved space-time In particular, the first derivation of the renormalization group in a physical mass-dependent scheme has been performed Unfortunately, the calculation was possible only in the framework of a linearized gravity approximation, that is why we did not obtain the physical β-functions for the cosmological and ewton constants The derivation of EA on a more complicated backgrounds in necessary for obtaining these β-functions, which can help us in order to prove or disprove the possibility of a time-dependent cosmological constant [6 The same calculation is vital for further theoretical development of the anomaly-induced inflation model [5 An important application of our calculation [7 of the effective action for massive scalar field is better understanding of the ambiguity in the conformal anomaly in n = 4, the issue which produced a long-standing doubts [7 We have seen that the dimensional regularization does not enable one to predict a coefficient α and that a similar ambiguity takes place in other calculational schemes, including taking a conformal limit in a massive theory and in the powerful Pauli-Villars regularization At the same time the ambiguity concerns only g R in the anomaly and the induced dynamics coming from the vacuum effective action can be made completely unambiguous by introducing the gr -term into classical action and imposing the corresponding renormalization condition at quantum level Acknowledgments The work of MA is partially supported by CICYT grant FPA and DGIID-DGA grant005-e4/ The work of EVG was supported by the atural Sciences and Engineering Research Council of Canada ISh is grateful to CPq and FAPEMIG and ICTP for permanent support and also to ICTP, Theory Group at CER and to Departamento de Física Teórica, Universidad de Zaragoza for kind hospitality [ D Birell and PCW Davies, Quantum Fields in Curved Space Cambridge Univ Press, Cambridge, 98 [ IL Buchbinder, SD Odintsov and IL Shapiro, Effective Action in Quantum Gravity IOP, Bristol, 99 [3 RJ Reigert, Phys Lett B 34, ; ES Fradkin and AA Tseytlin, Phys Lett B 34, [4 IL Shapiro and AG Jacksenaev, Phys Lett B 34, [5 S Deser, MJ Duff, and C Isham, ucl Phys B, [6 MJ Duff, ucl Phys B 5, [7 EV Gorbar, IL Shapiro, JHEP ; JHEP 06, [8 EV Gorbar, IL Shapiro, JHEP 0, [9 G de Berredo-Peixoto, EV Gorbar, and IL Shapiro, Class Quant Grav,, 8 004; [hep-th/039 [0 M Asorey, EV Gorbar, and IL Shapiro, Class Quant Grav, [ T Appelquist, J Carazzone, Phys Rev D, [ I G Avramidi, Yad Fiz Sov Journ ucl Phys 49, [3 AO Barvinsky, GA Vilkovisky, ucl Phys B 333, [4 IL Shapiro, Int Journal of Mod Phys D, 59 00; [Hep-ph/0038 [5 JC Fabris, AM Pelinson, and IL Shapiro, ucl Phys B 597, ; IL Shapiro, J Solà Phys Lett B 530, 0 00 AM Pelinson, IL Shapiro, and FI Takakura, ucl Phys B 648, [6 I Shapiro, J Solà, Phys Lett 475, ; JHEP 00, ; IL Shapiro, J Solà, C España-Bonet, and P Ruiz-Lapuente, Phys Lett B 574, ; IL Shapiro, J Solà, and H Stefancic, JCAP 050, [7 MJ Duff, Class Quant Grav, [8 SM Christensen, Phys Rev D 4, ; 7D, ; 4D, [9 SW Hawking, Comm Math Phys 55, [0 This statement must be taken with caution in the case of interacting fields [ [ SJ Hathrell, Ann Phys 39; 4 98 [ G de Berredo-Peixoto and IL Shapiro, Phys Lett B 54, ; [hep-th/0058 [3 AA Slavnov, Theor Math Phys 33, TD Bakeyev and AA Slavnov, Mod Phys Lett A, [4 M Asorey and F Falceto, ucl Phys B 37, ; Phys Rev D 54, M Asorey, F Falceto, J L López, and G Luzón, ucl Phys B 49, JL López, Universalidad y Regularización Ultravioleta en Teorías Gauge PhD dissertation, Zaragoza University, 995
Effective approach and decoupling in QFT
Effective approach and decoupling in QFT Ilya L. Shapiro Departamento de Física, Universidade Federal de Juiz de Fora, MG, Brazil Based on collaborations with M. Asorey (DFTUZ), Ed. Gorbar (BITP), J. Solà
More informationGravitational Waves and New Perspectives for Quantum Gravity
Gravitational Waves and New Perspectives for Quantum Gravity Ilya L. Shapiro Universidade Federal de Juiz de Fora, Minas Gerais, Brazil Supported by: CAPES, CNPq, FAPEMIG, ICTP Challenges for New Physics
More informationManifestly diffeomorphism invariant classical Exact Renormalization Group
Manifestly diffeomorphism invariant classical Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for Asymptotic Safety seminar,
More informationQuantum Fields in Curved Spacetime
Quantum Fields in Curved Spacetime Lecture 3 Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 22, 2016. Recap AdS 3 is an instructive application of quantum fields in curved space. The
More informationTowards a manifestly diffeomorphism invariant Exact Renormalization Group
Towards a manifestly diffeomorphism invariant Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for UK QFT-V, University of Nottingham,
More informationRegularization Physics 230A, Spring 2007, Hitoshi Murayama
Regularization Physics 3A, Spring 7, Hitoshi Murayama Introduction In quantum field theories, we encounter many apparent divergences. Of course all physical quantities are finite, and therefore divergences
More informationTheory of Elementary Particles homework VIII (June 04)
Theory of Elementary Particles homework VIII June 4) At the head of your report, please write your name, student ID number and a list of problems that you worked on in a report like II-1, II-3, IV- ).
More informationNew Model of massive spin-2 particle
New Model of massive spin-2 particle Based on Phys.Rev. D90 (2014) 043006, Y.O, S. Akagi, S. Nojiri Phys.Rev. D90 (2014) 123013, S. Akagi, Y.O, S. Nojiri Yuichi Ohara QG lab. Nagoya univ. Introduction
More informationStatus of Hořava Gravity
Status of Institut d Astrophysique de Paris based on DV & T. P. Sotiriou, PRD 85, 064003 (2012) [arxiv:1112.3385 [hep-th]] DV & T. P. Sotiriou, JPCS 453, 012022 (2013) [arxiv:1212.4402 [hep-th]] DV, arxiv:1502.06607
More informationConformal factor dynamics in the 1=N. expansion. E. Elizalde. Department ofphysics, Faculty of Science, Hiroshima University. and. S.D.
Conformal factor dynamics in the =N expansion E. Elizalde Department ofphysics, Faculty of Science, Hiroshima University Higashi-Hiroshima 74, Japan and S.D. Odintsov Department ECM, Faculty ofphysics,
More informationHIGHER SPIN PROBLEM IN FIELD THEORY
HIGHER SPIN PROBLEM IN FIELD THEORY I.L. Buchbinder Tomsk I.L. Buchbinder (Tomsk) HIGHER SPIN PROBLEM IN FIELD THEORY Wroclaw, April, 2011 1 / 27 Aims Brief non-expert non-technical review of some old
More informationQuantum Effects in Softly Broken Gauge Theories in Curved Space-Times
Quantum Effects in Softly Broken Gauge Theories in Curved Space-Times I.L. Buchbinder a,b1, G. de Berredo-Peixoto b 2, I.L. Shapiro b 3 (a) Department of Theoretical Physics, Tomsk State Pedagogical University
More informationarxiv: v2 [gr-qc] 2 Feb 2011
Dynamics of the Laplace-Runge-Lenz vector in the quantum-corrected Newton gravity C. Farina a, W. J. M. Kort-Kamp a, Sebastiao Mauro b, Ilya L. Shapiro b1 (a) Instituto de Física, Universidade Federal
More informationRenormalizability in (noncommutative) field theories
Renormalizability in (noncommutative) field theories LIPN in collaboration with: A. de Goursac, R. Gurău, T. Krajewski, D. Kreimer, J. Magnen, V. Rivasseau, F. Vignes-Tourneret, P. Vitale, J.-C. Wallet,
More information1 Running and matching of the QED coupling constant
Quantum Field Theory-II UZH and ETH, FS-6 Assistants: A. Greljo, D. Marzocca, J. Shapiro http://www.physik.uzh.ch/lectures/qft/ Problem Set n. 8 Prof. G. Isidori Due: -5-6 Running and matching of the QED
More informationGraviton contributions to the graviton self-energy at one loop order during inflation
Graviton contributions to the graviton self-energy at one loop order during inflation PEDRO J. MORA DEPARTMENT OF PHYSICS UNIVERSITY OF FLORIDA PASI2012 1. Description of my thesis problem. i. Graviton
More informationA Brief Introduction to AdS/CFT Correspondence
Department of Physics Universidad de los Andes Bogota, Colombia 2011 Outline of the Talk Outline of the Talk Introduction Outline of the Talk Introduction Motivation Outline of the Talk Introduction Motivation
More informationarxiv:hep-th/ v1 2 May 1997
Exact Renormalization Group and Running Newtonian Coupling in Higher Derivative Gravity arxiv:hep-th/9705008v 2 May 997 A.A. Bytseno State Technical University, St. Petersburg, 95252, Russia L.N. Granda
More informationOne Loop Tests of Higher Spin AdS/CFT
One Loop Tests of Higher Spin AdS/CFT Simone Giombi UNC-Chapel Hill, Jan. 30 2014 Based on 1308.2337 with I. Klebanov and 1401.0825 with I. Klebanov and B. Safdi Massless higher spins Consistent interactions
More informationVacuum Energy and Effective Potentials
Vacuum Energy and Effective Potentials Quantum field theories have badly divergent vacuum energies. In perturbation theory, the leading term is the net zero-point energy E zeropoint = particle species
More informationPlenty of Nothing: Black Hole Entropy in Induced Gravity
J. Astrophys. Astr. (1999) 20, 121 129 Plenty of Nothing: Black Hole Entropy in Induced Gravity V. P. Frolov, Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Canada
More informationNTNU Trondheim, Institutt for fysikk
FY3464 Quantum Field Theory II Final exam 0..0 NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory II Contact: Kåre Olaussen, tel. 735 9365/4543770 Allowed tools: mathematical
More informationWhy we need quantum gravity and why we don t have it
Why we need quantum gravity and why we don t have it Steve Carlip UC Davis Quantum Gravity: Physics and Philosophy IHES, Bures-sur-Yvette October 2017 The first appearance of quantum gravity Einstein 1916:
More informationStress-energy tensor is the most important object in a field theory and have been studied
Chapter 1 Introduction Stress-energy tensor is the most important object in a field theory and have been studied extensively [1-6]. In particular, the finiteness of stress-energy tensor has received great
More informationNon-locality in QFT due to Quantum Effects in Gravity
Non-locality in QFT due to Quantum Effects in Gravity Xavier Calmet Physics & Astronomy University of Sussex 1 Effective action for GR How can we describe general relativity quantum mechanically? Well
More informationarxiv:astro-ph/ v4 4 Sep 2003
Variable Cosmological Constant as a Planck Scale Effect Ilya L. Shapiro, 1 Joan Solà 2 Departamento de Física, ICE, Universidade Federal de Juiz de Fora, MG, Brazil Juiz de Fora, 36036-330, Minas Gerais,
More informationQuantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University
Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1
More informationSome Quantum Aspects of D=3 Space-Time Massive Gravity.
Some Quantum Aspects of D=3 Space-Time Massive Gravity. arxiv:gr-qc/96049v 0 Nov 996 Carlos Pinheiro, Universidade Federal do Espírito Santo, Departamento de Física, Vitória-ES, Brazil, Gentil O. Pires,
More informationEspansione a grandi N per la gravità e 'softening' ultravioletto
Espansione a grandi N per la gravità e 'softening' ultravioletto Fabrizio Canfora CECS Valdivia, Cile Departimento di fisica E.R. Caianiello NFN, gruppo V, CG Salerno http://www.sa.infn.it/cqg , Outline
More informationDirac Equation with Self Interaction Induced by Torsion
Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 12, 587-594 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.5773 Dirac Equation with Self Interaction Induced by Torsion Antonio
More informationarxiv: v3 [hep-th] 30 May 2018
arxiv:1010.0246v3 [hep-th] 30 May 2018 Graviton mass and cosmological constant: a toy model Dimitrios Metaxas Department of Physics, National Technical University of Athens, Zografou Campus, 15780 Athens,
More informationEffective Field Theory in Cosmology
C.P. Burgess Effective Field Theory in Cosmology Clues for cosmology from fundamental physics Outline Motivation and Overview Effective field theories throughout physics Decoupling and naturalness issues
More informationConformal Transformation in Gravity. Ilya L. Shapiro 1. and. Tomsk Pedagogical Institute, Tomsk, Russia. Hiroyuki Takata 2. and.
Conformal Transformation in Gravity DFTU/95/4 March, 995 Ilya L. Shapiro Departamento de Fisica Teorica, Universidad de aragoza, 50009, aragoza, Spain and Tomsk Pedagogical Institute, 63404 Tomsk, Russia.
More informationThe ultraviolet behavior of quantum gravity with fakeons
The ultraviolet behavior of quantum gravity with fakeons Dipartimento di Fisica Enrico Fermi SW12: Hot Topics in Modern Cosmology Cargèse May 18 th, 2018 Outline 1 Introduction 2 Lee-Wick quantum eld theory
More informationarxiv:gr-qc/ v1 29 Nov 1994
QUANTUM COSMOLOGY FOR A QUADRATIC THEORY OF GRAVITY Luis O. Pimentel 1 and Octavio Obregón 1,2 arxiv:gr-qc/9411072v1 29 Nov 1994 1 Departamento de Física, Universidad Autónoma Metropolitana, Apartado Postal
More informationHiggs inflation: dark matter, detectability and unitarity
Higgs inflation: dark matter, detectability and unitarity Rose Lerner University of Helsinki and Helsinki Institute of Physics In collaboration with John McDonald (Lancaster University) 0909.0520 (Phys.
More informationSolutions to gauge hierarchy problem. SS 10, Uli Haisch
Solutions to gauge hierarchy problem SS 10, Uli Haisch 1 Quantum instability of Higgs mass So far we considered only at RGE of Higgs quartic coupling (dimensionless parameter). Higgs mass has a totally
More informationQuantum discontinuity between zero and infinitesimal graviton mass with a Λ term. Abstract
MCTP-01-06 hep-th/010093 Quantum discontinuity between zero and infinitesimal graviton mass with a Λ term F. A. Dilkes, M. J. Duff, James T. Liu and H. Sati Michigan Center for Theoretical Physics Randall
More informationarxiv:hep-ph/ v3 19 Oct 2001
A Note on Regularization methods in Kaluza-Klein Theories Roberto Contino, Luigi Pilo arxiv:hep-ph/0104130v3 19 Oct 2001 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy & INFN Abstract
More informationarxiv: v2 [gr-qc] 1 Oct 2009
On the cosmological effects of the Weyssenhoff spinning fluid in the Einstein-Cartan framework Guilherme de Berredo-Peixoto arxiv:0907.1701v2 [gr-qc] 1 Oct 2009 Departamento de Física, ICE, Universidade
More informationChern-Simons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee
Chern-Simons Theory and Its Applications The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee Maxwell Theory Maxwell Theory: Gauge Transformation and Invariance Gauss Law Charge Degrees of
More informationThe cosmological constant puzzle
The cosmological constant puzzle Steven Bass Cosmological constant puzzle: Accelerating Universe: believed to be driven by energy of nothing (vacuum) Vacuum energy density (cosmological constant or dark
More informationQuantum Gravity and the Renormalization Group
Nicolai Christiansen (ITP Heidelberg) Schladming Winter School 2013 Quantum Gravity and the Renormalization Group Partially based on: arxiv:1209.4038 [hep-th] (NC,Litim,Pawlowski,Rodigast) and work in
More informationA New Approach to the Cosmological Constant Problem, and Dark Energy. Gregory Gabadadze
A New Approach to the Cosmological Constant Problem, and Dark Energy Gregory Gabadadze New York University GG, Phys. Lett. B, 2014, and work in preparation with Siqing Yu October 12, 2015 Einstein s equations
More informationInflationary cosmology from higher-derivative gravity
Inflationary cosmology from higher-derivative gravity Sergey D. Odintsov ICREA and IEEC/ICE, Barcelona April 2015 REFERENCES R. Myrzakulov, S. Odintsov and L. Sebastiani, Inflationary universe from higher-derivative
More informationarxiv:hep-th/ v1 28 May 1996
NAMBU-JONA-LASINIO MODEL IN CURVED SPACETIME WITH MAGNETIC FIELD arxiv:hep-th/9605195v1 28 May 1996 B. Geyer 1 Institute of Theoretical Physics and Center for Theoretical Sciences Leipzig University, Augustusplatz
More informationQFT Dimensional Analysis
QFT Dimensional Analysis In the h = c = 1 units, all quantities are measured in units of energy to some power. For example m = p µ = E +1 while x µ = E 1 where m stands for the dimensionality of the mass
More informationOn particle production by classical backgrounds
On particle production by classical backgrounds arxiv:hep-th/0103251v2 30 Mar 2001 Hrvoje Nikolić Theoretical Physics Division, Rudjer Bošković Institute, P.O.B. 180, HR-10002 Zagreb, Croatia hrvoje@faust.irb.hr
More informationScattering Amplitudes
Scattering Amplitudes LECTURE 1 Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP), UC Davis ICTP Summer School, June 2017 Particle experiments: our probe to fundamental laws of Nature Theorist
More information2P + E = 3V 3 + 4V 4 (S.2) D = 4 E
PHY 396 L. Solutions for homework set #19. Problem 1a): Let us start with the superficial degree of divergence. Scalar QED is a purely bosonic theory where all propagators behave as 1/q at large momenta.
More informationQuantum Dynamics of Supergravity
Quantum Dynamics of Supergravity David Tong Work with Carl Turner Based on arxiv:1408.3418 Crete, September 2014 An Old Idea: Euclidean Quantum Gravity Z = Dg exp d 4 x g R topology A Preview of the Main
More informationSTANDARD MODEL and BEYOND: SUCCESSES and FAILURES of QFT. (Two lectures)
STANDARD MODEL and BEYOND: SUCCESSES and FAILURES of QFT (Two lectures) Lecture 1: Mass scales in particle physics - naturalness in QFT Lecture 2: Renormalisable or non-renormalisable effective electroweak
More informationMaxwell s equations. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationGravity vs Yang-Mills theory. Kirill Krasnov (Nottingham)
Gravity vs Yang-Mills theory Kirill Krasnov (Nottingham) This is a meeting about Planck scale The problem of quantum gravity Many models for physics at Planck scale This talk: attempt at re-evaluation
More informationDefining Chiral Gauge Theories Beyond Perturbation Theory
Defining Chiral Gauge Theories Beyond Perturbation Theory Lattice Regulating Chiral Gauge Theories Dorota M Grabowska UC Berkeley Work done with David B. Kaplan: Phys. Rev. Lett. 116 (2016), no. 21 211602
More informationLoop corrections in Yukawa theory based on S-51
Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed
More informationContact interactions in string theory and a reformulation of QED
Contact interactions in string theory and a reformulation of QED James Edwards QFT Seminar November 2014 Based on arxiv:1409.4948 [hep-th] and arxiv:1410.3288 [hep-th] Outline Introduction Worldline formalism
More informationGauge Theory of Gravitation: Electro-Gravity Mixing
Gauge Theory of Gravitation: Electro-Gravity Mixing E. Sánchez-Sastre 1,2, V. Aldaya 1,3 1 Instituto de Astrofisica de Andalucía, Granada, Spain 2 Email: sastre@iaa.es, es-sastre@hotmail.com 3 Email: valdaya@iaa.es
More informationEmergent Spacetime. Udit Gupta. May 14, 2018
Emergent Spacetime Udit Gupta May 14, 2018 Abstract There have been recent theoretical hints that spacetime should not be thought of as a fundamental concept but rather as an emergent property of an underlying
More informationNTNU Trondheim, Institutt for fysikk
NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 998971 Allowed tools: mathematical tables 1. Spin zero. Consider a real, scalar field
More informationTopologically Massive Gravity and AdS/CFT
Topologically Massive Gravity and AdS/CFT Institute for Theoretical Physics University of Amsterdam The Planck Scale, XXV Max Born Symposium Wroclaw, 30 June 2009 Introduction Three dimensional gravity
More informationHigher spins and twistor theory
Higher spins and twistor theory Tim Adamo Imperial College London New Horizons in Twistor Theory 5 January 2017 Work with P. Haehnel & T. McLoughlin [arxiv:1611.06200] T Adamo (Imperial) Higher spins +
More informationSymmetries, Groups Theory and Lie Algebras in Physics
Symmetries, Groups Theory and Lie Algebras in Physics M.M. Sheikh-Jabbari Symmetries have been the cornerstone of modern physics in the last century. Symmetries are used to classify solutions to physical
More informationZhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016
Zhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016 We are directly observing the history of the universe as we look deeply into the sky. JUN 30, 2016 ZZXianyu (CMSA) 2 At ~10 4 yrs the universe becomes
More informationAsymptotic safety of gravity and the Higgs boson mass. Mikhail Shaposhnikov Quarks 2010, Kolomna, June 6-12, 2010
Asymptotic safety of gravity and the Higgs boson mass Mikhail Shaposhnikov Quarks 2010, Kolomna, June 6-12, 2010 Based on: C. Wetterich, M. S., Phys. Lett. B683 (2010) 196 Quarks-2010, June 8, 2010 p.
More informationWhere are we heading? Nathan Seiberg IAS 2016
Where are we heading? Nathan Seiberg IAS 2016 Two half-talks A brief, broad brush status report of particle physics and what the future could be like The role of symmetries in physics and how it is changing
More informationQUANTUM FIELD THEORY. A Modern Introduction MICHIO KAKU. Department of Physics City College of the City University of New York
QUANTUM FIELD THEORY A Modern Introduction MICHIO KAKU Department of Physics City College of the City University of New York New York Oxford OXFORD UNIVERSITY PRESS 1993 Contents Quantum Fields and Renormalization
More informationInequivalence of First and Second Order Formulations in D=2 Gravity Models 1
BRX TH-386 Inequivalence of First and Second Order Formulations in D=2 Gravity Models 1 S. Deser Department of Physics Brandeis University, Waltham, MA 02254, USA The usual equivalence between the Palatini
More informationA Note On The Chern-Simons And Kodama Wavefunctions
hep-th/0306083 arxiv:gr-qc/0306083v2 19 Jun 2003 A Note On The Chern-Simons And Kodama Wavefunctions Edward Witten Institute For Advanced Study, Princeton NJ 08540 USA Yang-Mills theory in four dimensions
More informationStephen Blaha, Ph.D. M PubHsMtw
Quantum Big Bang Cosmology: Complex Space-time General Relativity, Quantum Coordinates,"Dodecahedral Universe, Inflation, and New Spin 0, 1 / 2,1 & 2 Tachyons & Imagyons Stephen Blaha, Ph.D. M PubHsMtw
More informationNonsingular big-bounce cosmology from spin and torsion
Nonsingular big-bounce cosmology from spin and torsion Nikodem J. Popławski Department of Physics, Indiana University, Bloomington, IN 22 nd Midwest Relativity Meeting University of Chicago, Chicago, IL
More informationHigher-derivative relativistic quantum gravity
Preprint-INR-TH-207-044 Higher-derivative relativistic quantum gravity S.A. Larin Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 732, Russia
More informationDark energy and nonlocal gravity
Dark energy and nonlocal gravity Michele Maggiore Vacuum 2015, Barcelona based on Jaccard, MM, Mitsou, PRD 2013, 1305.3034 MM, PRD 2014, 1307.3898 Foffa, MM, Mitsou, PLB 2014, 1311.3421 Foffa, MM, Mitsou,
More informationBPS Solitons and Killing Spinors in Three Dimensional N =2Supergravity
La Plata-Th 97/18 BPS Solitons and Killing Spinors in Three Dimensional N =2Supergravity José D. Edelstein Departamento de Física, Universidad Nacional de La Plata C.C. 67, (1900) La Plata, Argentina Short
More informationUniversal invariant renormalization of supersymmetric Yang-Mills theory.
Universal invariant renormalization of supersymmetric Yang-Mills theory. arxiv:hep-th/0305128v1 15 May 2003 A.A.Slavnov Steklov Mathematical Institute, 117966, Gubkina, 8, Moscow, Russia and Moscow State
More informationGravitational Waves. GR: 2 polarizations
Gravitational Waves GR: 2 polarizations Gravitational Waves GR: 2 polarizations In principle GW could have 4 other polarizations 2 vectors 2 scalars Potential 4 `new polarizations Massive Gravity When
More information+ µ 2 ) H (m 2 H 2
I. THE HIGGS POTENTIAL AND THE LIGHT HIGGS BOSON In the previous chapter, it was demonstrated that a negative mass squared in the Higgs potential is generated radiatively for a large range of boundary
More informationarxiv:hep-th/ v1 17 Jan 2007
Generating mass and topological terms to the antisymmetric tensor matter field by Higgs mechanism arxiv:hep-th/0701161v1 17 Jan 2007 L. Gonzaga Filho, M. S. Cunha, a C. A. S. Almeida and R. R. Landim b,1
More informationQuasilocal vs Holographic Stress Tensor in AdS Gravity
Quasilocal vs Holographic Stress Tensor in AdS Gravity Rodrigo Olea Universidad Andrés Bello Quantum Gravity in the Southern Cone VI Maresias, Sept 11-14, 2013 Rodrigo Olea (UNAB) () Quasilocal vs Holographic
More informationarxiv:gr-qc/ v2 21 Mar 2000
Conformal symmetry, anomaly and effective action for metric-scalar gravity with torsion arxiv:gr-qc/990708v2 2 Mar 2000 J.A. Helayel-Neto Centro Brasileiro de Pesquisas Físicas-CBPF-CNPq Universidade Católica
More informationarxiv:hep-th/ v1 7 Jun 1994
FTUAM 94/8 NIKHEF-H 94/14 Shift versus no-shift in local regularizations of Chern-Simons theory UPRF 94/395 arxiv:hep-th/9406034v1 7 Jun 1994 G. Giavarini Libera Università della Bassa Ovest, Villa Baroni
More information19 Entanglement Entropy in Quantum Field Theory
19 Entanglement Entropy in Quantum Field Theory So far we have discussed entanglement in ordinary quantum mechanics, where the Hilbert space of a finite region is finite dimensional. Now we will discuss
More informationQFT Dimensional Analysis
QFT Dimensional Analysis In h = c = 1 units, all quantities are measured in units of energy to some power. For example m = p µ = E +1 while x µ = E 1 where m stands for the dimensionality of the mass rather
More informationWhy Be Natural? Jonathan Bain. Department of Technology, Culture and Society Tandon School of Engineering, New York University Brooklyn, New York
Why Be Natural? Jonathan Bain Department of Technology, Culture and Society Tandon School of Engineering, New York University Brooklyn, New York 1. How to Construct an EFT. 2. Why Be Natural? - Modest
More informationHigher-Spin Fermionic Gauge Fields & Their Electromagnetic Coupling
Higher-Spin Fermionic Gauge Fields & Their Electromagnetic Coupling Rakibur Rahman Université Libre de Bruxelles, Belgium April 18, 2012 ESI Workshop on Higher Spin Gravity Erwin Schrödinger Institute,
More informationNON-ANALYTICITIES IN THREE-DIMENSIONAL GAUGE THEORIES
NON-ANALYTICITIES IN THREE-DIMENSIONAL GAUGE THEORIES M. ASOREY, D. GARCIA-ALVAREZ Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain. J.L. LÓPEZ Departamento de Matemáticas
More informationXV Mexican Workshop on Particles and Fields
XV Mexican Workshop on Particles and Fields Constructing Scalar-Photon Three Point Vertex in Massless Quenched Scalar QED Dra. Yajaira Concha Sánchez, Michoacana University, México 2-6 November 2015 Mazatlán,
More informationScale symmetry a link from quantum gravity to cosmology
Scale symmetry a link from quantum gravity to cosmology scale symmetry fluctuations induce running couplings violation of scale symmetry well known in QCD or standard model Fixed Points Quantum scale symmetry
More informationEffective Field Theory in Cosmology
C.P. Burgess Effective Field Theory in Cosmology Clues for cosmology from fundamental physics Outline Motivation and Overview Effective field theories throughout physics Decoupling and naturalness issues
More informationChris Verhaaren Joint Theory Seminar 31 October With Zackaria Chacko, Rashmish Mishra, and Simon Riquelme
Chris Verhaaren Joint Theory Seminar 31 October 2016 With Zackaria Chacko, Rashmish Mishra, and Simon Riquelme It s Halloween A time for exhibiting what some find frightening And seeing that it s not so
More informationAdS/CFT duality. Agnese Bissi. March 26, Fundamental Problems in Quantum Physics Erice. Mathematical Institute University of Oxford
AdS/CFT duality Agnese Bissi Mathematical Institute University of Oxford March 26, 2015 Fundamental Problems in Quantum Physics Erice What is it about? AdS=Anti de Sitter Maximally symmetric solution of
More informationOn Torsion Fields in Higher Derivative Quantum Gravity
Annales de la Fondation Louis de Broglie, Volume 3 no -3, 007 33 On Torsion Fields in Higher Derivative Quantum Gravity S.I. Kruglov University of Toronto at Scarborough, Physical and Environmental Sciences
More informationRenormalisation Group Flows in Four Dimensions and the a-theorem
Renormalisation Group Flows in Four Dimensions and the a-theorem John Cardy University of Oxford Oxford, January 2012 Renormalisation Group The RG, as applied to fluctuating systems extended in space or
More informationNon-Abelian tensor multiplet in four dimensions
PASCOS 2012 18th nternational Symposium on Particles Strings and Cosmology OP Publishing Non-Abelian tensor multiplet in four dimensions Hitoshi Nishino and Subhash Rajpoot, Department of Physics and Astronomy,
More informationHeisenberg-Euler effective lagrangians
Heisenberg-Euler effective lagrangians Appunti per il corso di Fisica eorica 7/8 3.5.8 Fiorenzo Bastianelli We derive here effective lagrangians for the electromagnetic field induced by a loop of charged
More informationAnomalies, Conformal Manifolds, and Spheres
Anomalies, Conformal Manifolds, and Spheres Nathan Seiberg Institute for Advanced Study Jaume Gomis, Po-Shen Hsin, Zohar Komargodski, Adam Schwimmer, NS, Stefan Theisen, arxiv:1509.08511 CFT Sphere partition
More informationIf I only had a Brane
If I only had a Brane A Story about Gravity and QCD. on 20 slides and in 40 minutes. AdS/CFT correspondence = Anti de Sitter / Conformal field theory correspondence. Chapter 1: String Theory in a nutshell.
More informationarxiv:hep-th/ v2 13 Sep 2001
Compactification of gauge theories and the gauge invariance of massive modes. Amorim a and J. Barcelos-Neto b Instituto de Física Universidade Federal do io de Janeiro J 21945-97 - Caixa Postal 68528 -
More informationHigher Spin AdS/CFT at One Loop
Higher Spin AdS/CFT at One Loop Simone Giombi Higher Spin Theories Workshop Penn State U., Aug. 28 2015 Based mainly on: SG, I. Klebanov, arxiv: 1308.2337 SG, I. Klebanov, B. Safdi, arxiv: 1401.0825 SG,
More informationOutline. 1 Relativistic field theory with variable space-time. 3 Extended Hamiltonians in field theory. 4 Extended canonical transformations
Outline General Relativity from Basic Principles General Relativity as an Extended Canonical Gauge Theory Jürgen Struckmeier GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany j.struckmeier@gsi.de,
More information